The pseudospectral method: Comparisons with finite differences for the elastic wave equation

Geophysics ◽  
1987 ◽  
Vol 52 (4) ◽  
pp. 483-501 ◽  
Author(s):  
Bengt Fornberg
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Difference Scheme ◽  
Finite Differences ◽  
Finite Difference Scheme ◽  
Pseudospectral Method ◽  
Two Dimensions ◽  
Seismic Modeling ◽  
Elastic Wave Equation

The pseudospectral (or Fourier) method has been used recently by several investigators for forward seismic modeling. The method is introduced here in two different ways: as a limit of finite differences of increasing orders, and by trigonometric interpolation. An argument based on spectral analysis of a model equation shows that the pseudospectral method (for the accuracies and integration times typical of forward elastic seismic modeling) may require, in each space dimension, as little as a quarter the number of grid points compared to a fourth‐order finite‐difference scheme and one‐sixteenth the number of points as a second‐order finite‐difference scheme. For the total number of points in two dimensions, these factors become 1/16 and 1/256, respectively; in three dimensions, they become 1/64 and 1/4 096, respectively. In a series of test calculations on the two‐dimensional elastic wave equation, only minor degradations are found in cases with variable coefficients and discontinuous interfaces.

scholarly journals ONE-STEP EXTRAPOLATION METHOD USING CHEBYSHEV’S RECURRENCE RELATION APPLIED TO SEISMIC MODELING

2016 ◽  
Vol 34 (4) ◽  
Author(s):  
Laura Lara Ortiz ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Difference Scheme ◽  
Chebyshev Polynomials ◽  
Finite Difference Scheme ◽  
Acoustic Wave Equation ◽  
Seismic Modeling ◽  
First Order ◽  
One Step

ABSTRACT. In this work we show that the solution of the first order differential wave equation for an analytical wavefield, using a finite-difference scheme in time, follows exactly the same recursion of modified Chebyshev polynomials. Based on this, we proposed a numerical...Keywords: seismic modeling, acoustic wave equation, analytical wavefield, Chebyshev polinomials. RESUMO. Neste trabalho, mostra-se que a solução da equação de onda de primeira ordem com um campo de onda analítico usando um esquema de diferenças finitas no tempo segue exatamente a relação de recorrência dos polinômios modificados de Chebyshev. O algoritmo...Palavras-chave: modelagem sísmica, equação da onda acústica, campo analítico, polinômios de Chebyshev.

Absorbing boundary condition for the elastic wave equation

Geophysics ◽  
1988 ◽  
Vol 53 (5) ◽  
pp. 611-624 ◽  
Author(s):  
C. J. Randall
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Grazing Incidence ◽  
Scalar Fields ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary Condition ◽  
Elastic Wave Equation ◽  
Absorbing Boundary

Extant absorbing boundary conditions for the elastic wave equation are generally effective only for waves nearly normally incident upon the boundary. High reflectivity is exhibited for waves traveling obliquely to the boundary. In this paper, a new and efficient absorbing boundary condition for two‐dimensional and three‐dimensional finite‐difference calculations of elastic wave propagation is presented. Compressional and shear components of the incident vector displacement fields are separated by calculating intermediary scalar potentials, allowing the use of Lindman’s boundary condition for scalar fields, which is highly absorbing for waves incident at any angle. The elastic medium is assumed to be homogeneous in the region immediately adjacent to the boundary. The reflectivity matrix of the resulting absorbing boundary for elastic waves is calculated, including the effects of finite‐difference truncation error. For effectively all angles of incidence, reflectivities are much smaller than those of the commonly employed paraxial absorbing boundaries, and the boundary condition is stable for any physical Poisson’s ratio. The nearly complete absorption predicted by the reflectivity matrix calculations, even at near grazing incidence, is demonstrated in a finite‐difference application.

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